Rotations and Cyclic Derivatives of Words in Unitary Letters

Let $\mathbb{N}$ be the natural numbers, for $k\in\mathbb{N}$ let $[k]$ be the initial segment determined by $k$, and let $d\in\mathbb{N}$. Let $W_{2d}$, with concatenation $(u,v)\mapsto uv$, and the generators $g(l)$ and signs $\varepsilon(l)$ of letters be as in Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words; for a word $w$ of length $k$ and $a,b\in[k]$ with $a\le b$, $w_{[a,b]}$ is the word $w_{a}w_{a+1}\cdots w_{b}$ of The Free Unitary Heat Generator on Unitary Laws. Let word polynomials, the word polynomials $e_{w}$, and pointwise sums and complex multiples be as in Word Polynomials in Unitary Letters: the Normed *-Algebra of Finitely Supported Word Functions; a sum of finitely many word polynomials is a word polynomial, with the union of their support sets as support set, finite by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §finite-union, and a sum over the empty index set is the zero word polynomial $0$, which has every nonempty finite subset of $W_{2d}$ as a support set. Let $\mathbb{C}$ be the complex numbers with imaginary unit $\mathrm{i}$.

1. (Rotations)¶ Let $w$ be a word of length $k\in\mathbb{N}$ and $m\in[k]$. If $\varepsilon(w_{m})=1$, the rotation of $w$ at $m$ is $r_{m}(w)=w$ when $m=1$ and $r_{m}(w)=w_{[m,k]}\,w_{[1,m-1]}$ when $1<m$. If $\varepsilon(w_{m})=-1$, it is $r_{m}(w)=w$ when $m=k$ and $r_{m}(w)=w_{[m+1,k]}\,w_{[1,m]}$ when $m<k$. Here $m-1$ and $m+1$ lie in $[k]$ in the respective cases, as in The Free Unitary Heat Generator on Unitary Laws.

2. (Cyclic derivatives)¶ For $j\in[d]$, the $j$-th cyclic derivative of $\varnothing$ is $D^{j}_{\varnothing}=0$, and that of a word $w$ of length $k\in\mathbb{N}$ is the word polynomial

$$D^{j}_{w}=\sum_{m\in[k],\ g(w_{m})=j}\mathrm{i}\,\varepsilon(w_{m})\,e_{r_{m}(w)}.$$